LAUSR

Abstract This paper studies extreme response statistics of random vibrations for a Jeffcott-type rotor with non-linear restoring force, under uniaxial colored noise excitation. The latter qqtype of dynamic system is of wide use in stability studies of rotating machinery, especially in the automotive industry. System response statistics are studied by applying the path integration (PI) method, which is based on the Markov property of the response process. The response statistics of the Jeffcott rotor are then obtained by solving the Fokker–Planck–Kolmogorov equation for a 6D (six dimensional) dynamic system. The resulting response probability distributions can serve as an engineering tool for determining a wide range of design issues, e.g. characteristic values, extreme value statistics, system reliability and first passage probability. Assessment of transverse random vibrations of shafts in rotating machinery may be of practical importance for applications with substantial environmental dynamic loads on supports, particularly in transport/vehicle engineering. Colored noise is a step forward compared to white noise excitation forces, but it raises the mechanical system dimension from 4D (four dimensional) to 6D. The major advantage of path integration, relative to direct Monte Carlo simulation, is that path integration yields high accuracy in the probability distribution tail. Moreover, it has better potential for pushing system parameters beyond dynamic system stability limits, affecting the numerical solution. Improved implementation of the PI algorithm was applied, specifically the fast Fourier transform (FFT) was used to simulate the dynamic system additive noise. PI was accelerated by using a Monte Carlo estimated joint probability density function as an initial input. Finally, the key feature of this work is the advance to 6D problems, where very little PI research has been done. The obvious reason is of course the formidable computation load arising from fine 6D meshes. This paper, however, gives a practical solution to the latter challenge, enabling 6D PI calculation on an ordinary desktop within a reasonable amount of time. Using modern computational hardware with a decent GPU (Graphic Processing Unit) will significantly facilitate 6D calculation, making it an easy engineering task. Note that most of the previous PI studies where done exclusively on CPU.